Financial Mathematics

Identify balances, flows and costs

A balance is an amount at a particular time. A cash flow is money entering or leaving at a specified time or over a period. Income of $1500 per month is a rate of inflow; a savings balance of $1500 is a stock. The matching numbers do not make them the same quantity.

A principal is an initial amount invested or borrowed. Interest is an amount charged or earned under an agreed rate and calculation method. Fees can add costs that are not described by the interest formula alone. The Financial Consumer Agency of Canada’s loan guidance emphasises regular payments, term, interest and applicable fees when considering total costs.

Use consistent signs from one person’s perspective: money received is positive and money paid is negative, or vice versa if stated. Never reverse the sign convention halfway through a timeline.

Reconcile a monthly plan without double counting

Consider a hypothetical monthly cash plan with $1500 received, $700 of fixed payments, $420 of variable spending and a $200 transfer to savings. The unallocated cash is

The savings transfer reduces the spending account but does not disappear from the person’s assets. Calling it both spending and a second loss of wealth would double count it. A budget concerns cash availability; a statement of assets and liabilities concerns what is owned and owed at a time.

A $420 weekly income has a 52-week planning total of $21 840. Dividing by 12 gives an average of $1820 per month under that convention, not $1680. Four weeks is not the same as an average calendar month, and an actual cash-flow calendar still depends on pay dates.

Use multiplicative factors for discounts and increases

A 15% reduction of an $80 price gives $80(0.85)=68$ dollars. Two successive reductions of 20% and 10% give

That is a 28% total reduction, not 30%, because the second reduction applies to the reduced price. A fee added afterwards changes the final outflow again.

Percentage changes need a reference. Recovering from a 20% decrease requires a 25% increase because $0.8\times1.25=1$. This asymmetry also matters when interpreting percentage returns. The general arithmetic is developed in Percentage.

Compare payment totals, then compare their timing

Suppose a purchase can be paid as $1200 now or four payments of $315 at the ends of months 1 through 4, with no other charges. The instalment total is $1260, which is $60 more in nominal dollars.

The timing is different. To compare at month zero under a stipulated monthly discount rate of 1%, convert every payment to that date:

Under that particular comparison rate, the present value of the instalments exceeds $1200 by about $29.12. At another comparison rate, the result changes. A present-value calculation does not by itself establish affordability, credit suitability or the person’s best choice.

To compare future payments with a present amount, state the discount rate and payment dates.

Distinguish rate labels before using a formula

An interest quotation should identify its time period and compounding rule. A 6% nominal annual rate compounded monthly corresponds to 0.5% per month in that specified model. A 6% effective annual rate instead has monthly equivalent $(1.06)^{1/12}-1$, not 0.5%.

A disclosed annual percentage rate or other cost-of-borrowing measure may include additional contractual elements. Do not automatically substitute it for the periodic interest rate in a balance recurrence. Read the definition supplied with the product and identify fees separately.

FCAC’s savings-account guidance explains why compounding frequency and account terms matter. The detailed mathematics follows in Simple Interest, Compound Interest and Loans and Amortization.

Separate nominal growth from purchasing power

Suppose a balance grows by a stipulated 5% over a year while a relevant price index grows by 3%. The purchasing-power factor is

The real increase relative to that index is therefore about 1.94%, not exactly 2%. Subtracting the rates gives an approximation, while dividing their growth factors gives the exact modelled comparison.

A person’s spending pattern need not match the chosen index. Taxes, fees and fluctuating returns may also change the realised result. These are model inputs to identify, not details that a larger compound-interest exponent automatically captures.

Explain what a financial comparison leaves out

A smaller regular payment may result from a longer repayment period rather than a lower cost. A higher projected return may carry different uncertainty, restrictions or timing. Distinguish a guaranteed contractual cash flow from a forecast before performing arithmetic on them.

For a transparent calculation, write down the principal, rate convention, dates, payments, fees, rounding rule and the comparison question. Then calculate. A spreadsheet or calculator that silently assumes different payment dates can be consistently precise about the wrong model.

These pages teach that quantitative structure. They do not select products, predict markets or replace a review of the actual agreement.

Related topics

Begin with Simple Interest for growth on an unchanged principal, then Compound Interest for interest added to the balance. Loans and Amortization adds the effect of repeated repayments.